Consider the kernel of the Magnus representation of the Torelli group and the kernel of the Burau representation of the braid group. We prove that for and for the groups and have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of “Johnson-type” homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of , we do this with the assistance of a computer calculation.
"Infinite generation of the kernels of the Magnus and Burau representations." Algebr. Geom. Topol. 10 (2) 837 - 851, 2010. https://doi.org/10.2140/agt.2010.10.837